Step right up, step right up! Try your luck at our fabulous contest. Guess right and you win a new car! Guess wrong, and you have to take home a goat!
You’re given the choice of three doors: Behind one door is a car. Behind the other two, smelly goats! You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3. He shows you that door No. 3 has a goat. He then says to you, “Do you want to switch to door No. 2?” Is it to your advantage to switch your choice?
Many people want to stay with their choice. This is because their gut reaction told them to pick door No. 1 and there is no reason to switch now. In fact, some contestants may even be suspicious of the game show hosts intentions. How do they know he won’t trick them and have them switch to a door with a goat!? A mathematical analysis is needed if clear thinking is to prevail.
As in similar situations, we need to break down this problem in to cases. To simplify the analysis, we will assume the car is always behind Door No. 1. This is because picking any other door and then doing a case by case analysis would be symmetric no matter what the numberings.
Case 1 (Pick Door No. 1):
Suppose you picked the door with the car, Door No. 1. But you don’t know it yet. And worse, the game show host is now opens door No. 2 and shows you a goat. He suggests you switch to door No. 3. If you switch, you will get a goat and lose. If you stay, you win a new car! Hmm, sticking with your original choice seems ideal in this situation.
Case 2 (Pick Door No. 2):
Now you picked a door with a goat, yuk! The host opens door No. 3 and shows you a goat. If you switch to door No. 1, you win the car. If you stay, you “win” the goat, not a good plan.
Case 3 (Pick Door No. 3):
Last case, you again have a door with a goat. The host shows you door No. 2 with a goat. If you switch to door No. 3 you win the car. If you stay, sadly, you will go home with the goat. Now let’s evaluate the strategies.
Say you are convinced that your gut is always right, and you never change your door. In case 1 you won. But your paranoia caused you to lose out big time in case 2 and 3. Hence, your chance of winning with the “stay” strategy is 1/3.
What if you are an indecisive person and you always change your mind (and your door) whenever possible. In case 1, you would change away from the car and get a goat. But in cases 2 and 3 your indecisiveness gave you the victory. Hence, your chance of winning with the “change” strategy is 2/3.
We can see that, overall, the better strategy is to change to the other door. This should be clear by the analysis above, and if you don’t believe me, try it for yourself next time you are on a game show.
However, if you would indulge me for just a moment, I want to muse on an interpretation of these results. If you pick any door at random, your chance of getting the car is 1/3. Next, the host opens a door to show you a goat. The math above states that, just by opening that door, the game has radically changed. If you stay with your door, you have a 1/3 chance of winning, just like before. But if you switch doors, suddenly your chance of winning jumps to 2/3!
But think about it logically. If there are only 2 doors left, and one has a goat and one has a car, shouldn’t your probability be 50-50, 1/2 chance either if you stay with your door or if you switch?
This is where the mystery of the problem comes in. Most people (wrongly) assume that since there are 2 doors left and 1 car they have a 50% chance of winning. So they see no reason to switch doors. In reality however, switching increases their chances of winning from 33% to 67%. But you protest, originally all 3 doors just had a 33% chance of winning, what in the world is going on?
The best I can explain it is this. When the game show host opened a door, he removed that door from the game. The 33% probability that that door was a winner had to go somewhere. And it did. This probability “collapsed” into the probability of the other door, giving the “switch” door a probability of 67% compared with your current “stay” door probability of 33%. Mind blowing. If you want the car, switch your door. If you want a goat, just say nope 🙂